The arrangement of complementary cones, known as the Q-arrangement, is not arbitrary: each cone has \(n\)  neighbors where two cones are neighbors if and only if they have \(n-1\)  common generators. In fact, the cones are in correspondence with the vertices of an \(n\) -dimensional hypercube graph \(Q_n\)  where the neighboring relationship is represented by the adjacency of the vertices of \(Q_n\) .
Stitching together all the cones, one at a time, along their common \((n-1)\) -facet amounts to following the longest Hamiltonian cycle of \(Q_n\)  (of length \(2^n\) ).
Notice that the collection of simplices need not form a simplicial set [1]} as the intersections of simplices need not be reduced to lower dimensional simplices.
